On invariant measures of nonlinear Markov processes
The paper deals with the stochastic differential equation in \(R^ d\), \[ dX(t) = \biggl[ - AX(t) + f \bigl( X(t), u(t) \bigr) \biggr] dt + dW(t), \] where \(\mu(t)\) is the probability law of \(X(t)\), \(A\) is a \(d \times d\) matrix, \(f\) is a function from \(R^ d \times M_ 2 (R^ d)\) to \(R^ d\), where \(M_ 2 (R^ d)\) is the space of all probability measures on \(R^ d\) which have finite second moments. The solution \(X(t)\) of this equation is a nonlinear Markov process in the sense of McKean. The aim of the paper is to prove existence and uniqueness of an invariant measure. A probability measure \(\rho\) is said to be invariant if \(\langle \rho, L(\rho) \varphi \rangle = 0\) for all \(C^ \infty\)- functions \(\varphi\) with compact support in \(R^ d\), where \(L(\rho)\) is the infinitesimal generator associated to the coefficients of the differential equation, with \(\mu (t) = \rho\). The authors give preliminary abstract results of existence and uniqueness of invariant measures for general equations. Then they apply these results to the previous particular problem, under appropriate conditions on \(A\) and \(f\).
existence and uniqueness of an invariant measure
scientific article; zbMATH DE number 527687
scientific article